Fanography

A tool to visually study the geography of Fano 3-folds.

Identification
Fano variety 3-8

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup

Picard rank
3 (others)
$-\mathrm{K}_X^3$
24
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 0
0 3 3
0 0 0 15
0 0 0
0 0
0
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
15
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is rational.

This variety is the blowup of

• 2-24, in a curve of genus 0
• 2-34, in a curve of genus 0
Deformation theory
number of moduli
3

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}$ 1 0
$0$ 0 3
Period sequence

The following period sequences are associated to this Fano 3-fold:

GRDB
#112
Fanosearch
#85
Extremal contractions
Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup formula.

Structure of quantum cohomology
Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety
$\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^2$
bundle
$\mathcal{O}(0,1,2) \oplus \mathcal{O}(1,1,0)$

See the big table for more information.